In this article https://www.guitar-chords.org.uk/c-major9-chord.html
Why is the 5th missing in the first and second shape, and the 3rd missing in the third shape?
Why are sometimes all the deegrees included and sometimes not?
Thanks
Yes, no and it depends. Technically you could name them differently but most of the time it depends on context. It might be that the chord in question “should” be played as the full chord but the guitarist either cannot physically play it or might just prefer the sound of a chord with one of more notes omitted and decide it doesn’t alter the song negatively.
You could name the chord differently if you wanted, i.e., C7 (no fifth) or something like that.
I can’t think of an example chord right now but there might be times where omitting a note results in the chord ending up with the same notes as another chord. It’s really up to you to some extent. Some chords do have their notes coincide but we often don’t think about it. For example, C6 and Am7 share the same notes. So do Fsus2 and Csus4, so it’s just about context.